A New Representation of Efficient Point Sets and Its Applications in DEA

Abstract

𝐸(𝑀,𝐾) (𝐸_𝑤(𝑀,𝐾)), the set of Pareto efficient (weak efficient) points of a set 𝑀 with respect to a cone 𝐾 in 𝑅^𝑛, is expressed as a differencebetweentwo sets 𝑀 and 𝑀+𝐾⧵{0} (𝑀 and 𝑀+int𝐾). Using the new representation, the properties of 𝐸(𝑀,𝐾) are proved more easily than before. When 𝑀 or 𝐾 is in the form of union, intersection, sum, or difference of two sets or two cones, respectively, the properties of 𝐸(𝑀,𝐾) are considered. Most of the properties are proved by the binary operations of sets, which is a new method in the multiobjective optimization. Then these properties are used to solve some types of multiobjective linear programming problems corresponding to Data Envelopment Analysis (DEA) models. The structures of the DEA efficient solution sets of four most representative DEA models are developed. Further more, the relationships between efficiencies of the four DEA models are deduced.

1. Introduction and Preliminaries

Multiobjective programming is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints (see [1–3]). Data Envelopment Analysis (DEA) is a nonparametric method in operations research and economics for the estimation of production frontiers. It is used to empirically measure productive efficiency of decision maker units DMU₁,…,DMU_𝑛 by solving the linear programming [4–6]. Charnes et al. and Wei et al. establish the equivalence of (weak) DEA efficient solutions in DEA model and (weak) Pareto solutions of multiobjective linear programming [7–9]. There is a multiobjective linear programming, corresponding to a DEA model, such that a DMU_𝑗0(1≤𝑗₀≤𝑛) is (weak) DEA efficient if and only if (𝑥_𝑗0,𝑦_𝑗0) (associating with the DMU_𝑗0) is a (weak) Pareto efficient solution of the multiobjective linear programming whose feasible region is the production possibility set (see [10]).

In this paper, we propose a new representation for the set of Pareto efficient (weak efficient) points. With the help of the new representation, not only the properties of the set of Pareto efficient (weak efficient) points which are given in [1, 3, 11] can be proved more simply, but also more new properties can be obtained. It is these new properties that reveal the relationships between the set of solutions and different multiobjective linear programmings which correspond to different DEA models. Further, the relationships between efficiencies of DMUs in different DEA models are obtained by a new way. Wei et al. [10] develop a famous method to translate production possibility sets in the intersection form and in the sum form and find all DEA efficient DMUs. For each of the four most representative DEA models, we offers a simple way to get all DEA efficient DMUs by the binary operations of sets.

Now let us recall the definition of efficiency and the representation of the set of efficient points deduced by the definition in vector optimization.

Definition 1 ([1], efficiency, weak efficiency). Given a nonempty set 𝑀 and a cone 𝐾 with int𝐾≠∅ in 𝑅^𝑛, 𝑥₀∈𝑀 is called a Pareto efficient (weak efficient) point of 𝑀, if there is no 𝑦∈𝑀 with 𝑦≠𝑥₀ such that 𝑥₀∈𝑦+𝐾(𝑥₀∈𝑦+int𝐾). The set of all Pareto efficient (weak efficient) points of 𝑀 is denoted by 𝐸(𝑀,𝐾)(𝐸_𝑤(𝑀,𝐾)).
𝐸(𝑀,𝐾) (𝐸_𝑤(𝑀,𝐾)) is called the efficient (weak efficient) point set of 𝑀. By Definition 1, we have that 𝐸()=,𝑀,𝐾𝑥∈𝑀∣thereisno𝑦∈𝑀,𝑦≠𝑥suchthat𝑥∈𝑦+𝐾𝐸_𝑤()=.𝑀,𝐾𝑥∈𝑀∣thereisno𝑦∈𝑀suchthat𝑥∈𝑦+int𝐾 (1)(2)
In [2] when 𝐾=𝑅^𝑛₊, the set is also described as follows: 𝐸(𝑀,𝐾)={𝑥∈𝑀∣𝑀∩(𝑥−𝐾)={𝑥}},𝐸_𝑤(𝑀,𝐾)={𝑥∈𝑀∣𝑀∩(𝑥−int𝐾)=∅}. (3)(4)
Since 𝐾=𝑅^𝑛₊ is a convex pointed cone, (1) and (3) are equivalent. So are (2) and (4). In Section 2, we give a new representation of the efficient (weak efficient) point set, which is expressed as the difference of two sets. The idea of the new representation is motivated by the following facts in the area of DEA:(i) the structures of the production possibility sets and the relationships between these sets (5);(ii) the equivalence of Pareto efficiency in multiobjective linear programming and DEA efficiency in DEA model (Theorem 18);(iii) the particularity of structures of the set of solutions to the multiobjective linear programmings corresponding to DEA models (detailed in Section 3).
For the four most representative DEA models 𝐶²𝑅, 𝐵𝐶², 𝐹𝐺, and 𝑆𝑇 (for the details about the models, see [4, 5]), each of the DEA models associates with a production possibility set which is also the feasible set of the multiobjective linear programming corresponding to this DEA model. The production possibility sets are denoted by 𝑇_𝐶²_𝑅, 𝑇_𝐵𝐶², 𝑇_𝐹𝐺, and, 𝑇_𝑆𝑇, respectively. The following relations hold (the structures of these production possibility sets are presented in Section 3):(a)𝑇_𝐶²_𝑅=𝑇_𝐹𝐺∪𝑇_𝑆𝑇,(b)𝑇_𝐵𝐶²=𝑇_𝐹𝐺∩𝑇_𝑆𝑇,(c)𝑇_𝐶²_𝑅⊃𝑇_𝐹𝐺⊃𝑇_𝐵𝐶²,(d)𝑇_𝐶²_𝑅⊃𝑇_𝑆𝑇⊃𝑇_𝐵𝐶². (5)
It is the specialty of the relations of the production possibility sets and the equivalence of DEA efficiency and Pareto efficiency that motivate us to propose a new representation of (weak) efficient point set. Using the new representation,we obtain some new properties of efficient point set 𝐸(𝑀,𝐾), when the set 𝑀 or 𝐾 is in form at union, intersection, sum, or difference of two sets. By these properties, it is easier to get the relationship between the DEA efficiency of the four DEA models than before.
This paper is organized as follows. Section 2 introduces the new representation of the efficient (weak efficient) point set of a set, discusses some new properties of efficient point set. Using the new expression of the set 𝐸(𝑀,𝐾)(𝐸_𝑤(𝑀,𝐾)), most of these properties are proved by the binary operations of sets. The multiobjective linear programming problems corresponding to the four DEA models are studied in Section 3. The structures of the efficient point sets and the efficient solution sets of the multiobjective linear programming problems are developed, and then the relationships between DEA efficiencies of DMUs in four DEA models are revealed. Section 4 is devoted to the conclusion.
The following notations are used in the paper.
Let 𝑀, 𝑀₁, 𝑀₂, and 𝐾 be sets in 𝑅^𝑛: 𝑀₁+𝑀₂=𝑥₁+𝑥₂∣𝑥₁∈𝑀₁,𝑥₂∈𝑀₂,𝑀₁⧵𝑀₂=𝑥∣𝑥∈𝑀₁,𝑥∉𝑀₂,𝑀=𝑥∈𝑅^𝑛,𝐾∣𝑥∉𝑀₀=𝐾⧵{0}. (6)

2. Some Properties of the Efficient Point Set

In this section, a new representation of 𝐸(𝑀,𝐾) (𝐸_𝑤(𝑀,𝐾)) is presented. Then we prove that it is equivalent to the original ones when 𝐾 is a cone. Lastly we focus on the properties of 𝐸(𝑀,𝐾), when 𝑀 or 𝐾 is in the form of the union, intersection, sum, or difference of two sets. Most of the proofs are completed by the binary operations of sets, which is a new method in multiobjective optimization.

Definition 2. Given a nonempty set 𝑀 and a cone 𝐾 with int𝐾≠∅ in 𝑅^𝑛, the efficient (weak efficient) point set of 𝑀 with respect to 𝐾 is defined by 𝐸(𝑀,𝐾)=𝑀⧵𝑀+𝐾₀,𝐸_𝑤(𝑀,𝐾)=𝑀⧵(𝑀+int𝐾). (7)(8)

Clearly, the following result holds.

Theorem 3. Equations (1) and (7) ((2) and (8)) are equivalent, that is, 𝐸(𝑀,𝐾)=𝑥∈𝑀𝑡ℎ𝑒𝑟𝑒𝑖𝑠𝑛𝑜𝑦∈𝑀,𝑦≠𝑥𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑥∈𝑦+𝐾=𝑀⧵𝑀+𝐾₀,𝐸_𝑤()=(.𝑀,𝐾𝑥∈𝑀∣𝑡ℎ𝑒𝑟𝑒𝑖𝑠𝑛𝑜𝑦∈𝑀𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑥∈𝑦+int𝐾=𝑀⧵𝑀+int𝐾)) (9)

Definition 2 gives a new representation of 𝐸(𝑀,𝐾) that is, the efficient point set of a set 𝑀 is the difference between two sets. Using this new representation, we prove the properties of 𝐸(𝑀,𝐾). Proposition 4 comes from Luc [1], Papageorgiou [3], and Guerraggio et al. [11]. We give an easier proof of this proposition.

Proposition 4. Assume that 𝑀, 𝑀₁, and 𝑀₂ are nonempty sets and 𝐾, 𝐾₁, and 𝐾₂ are cones in 𝑅^𝑛, 𝜆>0. Then(i)𝐸(𝑀,𝐾₂)⊆𝐸(𝑀,𝐾₁), if 𝐾₁⊆𝐾₂,(ii)𝐸(𝜆𝑀,𝐾)=𝜆𝐸(𝑀,𝐾),(iii)𝐸(𝑀₁+𝑀₂,𝐾)⊆𝐸(𝑀₁,𝐾)+𝐸(𝑀₂,𝐾),(iv)𝐸(𝑀+𝐻,𝐾)=𝐸(𝑀,𝐾), for any {0}⊆𝐻⊆𝐾, 𝐾   is a convex cone.

Proof. For convenience, let 𝐾₁₀=𝐾₁⧵{0}, and let 𝐾₂₀=𝐾₂⧵{0}. Since 𝐾₁⊆𝐾₂, we have 𝑀+𝐾₁₀⊆𝑀+𝐾₂₀. And then 𝐸𝑀,𝐾₂=𝑀⧵𝑀+𝐾₂₀⊆𝑀⧵𝑀+𝐾₁₀=𝐸𝑀,𝐾₁, (10)𝐸(𝜆𝑀,𝐾)=(𝜆𝑀)⧵𝜆𝑀+𝐾₀=(𝜆𝑀)⧵𝜆𝑀+𝜆𝐾₀=(𝜆𝑀)⧵𝜆𝑀+𝐾₀=𝜆𝑀⧵𝑀+𝐾₀=𝜆[𝐸(𝑀,𝐾)]. (11)
By Theorem 3, it is sufficient to show that 𝑀₁+𝑀₂⧵𝑀₁+𝑀₂+𝐾₀⊆𝑀₁⧵𝑀₁+𝐾₀+𝑀₂⧵𝑀₂+𝐾₀, (12) for all 𝑥=𝑥₁+𝑥₂∈(𝑀₁+𝑀₂)⧵(𝑀₁+𝑀₂+𝐾₀), where 𝑥₁∈𝑀₁, 𝑥₂∈𝑀₂ and 𝑥₁+𝑥₂∉𝑀₁+𝑀₂+𝐾₀. Then 𝑥₁∉𝑀₁+𝐾₀, otherwise 𝑥₁+𝑥₂∈𝑀₁+𝑀₂+𝐾₀, a contradiction. Similarly 𝑥₂∉𝑀₂+𝐾₀. Hence 𝑥=𝑥₁+𝑥₂∈𝑀₁⧵𝑀₁+𝐾₀+𝑀₂⧵𝑀₂+𝐾₀. (13)
Since 0∈𝐻⊆𝐾 and 𝐾 is a convex cone, we have 𝑀⊂𝑀+𝐻, 𝐻+𝐾₀=𝐾₀, 𝐸(𝑀+𝐻,𝐾)=(𝑀+𝐻)⧵𝑀+𝐻+𝐾₀=(𝑀+𝐻)⧵𝑀+𝐾₀⊃𝑀⧵𝑀+𝐾₀=𝐸(𝑀,𝐾). (14)
On the other hand, For any 𝑥∈𝐸(𝑀+𝐻,𝐾)=(𝑀+𝐻)⧵(𝑀+𝐾₀), 𝑥=𝑚+ℎ, 𝑚∈𝑀, ℎ∈𝐻. If ℎ≠0, then 𝑥∈(𝑀+𝐾₀), a contradiction. Hence, 𝑥∈𝑀, 𝑥∉𝑀+𝐾₀, that is, 𝑥∈𝐸(𝑀,𝐾).

A spacial case of (ii) in Proposition 4 is that 𝐸𝑀+⋯^𝑚+𝑀,𝐾=𝐸(𝑚𝑀,𝐾)=𝑚𝐸(𝑀,𝐾), (15) if 𝑀 is convex and 𝐾 is a cone.

Besides Proposition 4, we state the following properties.

Corollary 5. Consider the following:(v)𝐸𝑀,𝐾₁+𝐾₂⊆𝐸𝑀,𝐾₁∩𝐸𝑀,𝐾₂,(vi)𝐸𝑀,𝐾₁∩𝐾₂⊇𝐸𝑀,𝐾₁∪𝐸𝑀,𝐾₂. (16)

Proposition 6. Let 𝑀, 𝐺₁, and 𝐺₂ be sets in 𝑅^𝑛. Then 𝑀⧵𝑀+𝐺₁∪𝐺₂₀=𝑀⧵𝑀+𝐺₁₀∩𝑀⧵𝑀+𝐺₂₀. (17)

Proof. It is obvious that (𝐺₁∪𝐺₂)₀=𝐺₁₀∪𝐺₂₀ and 𝑀+(𝐺₁∪𝐺₂)₀=𝑀+(𝐺₁₀∪𝐺₂₀)=(𝑀+𝐺₁₀)∪(𝑀+𝐺₂₀). Hence 𝑀⧵𝑀+𝐺₁∪𝐺₂₀=𝑀⧵𝑀+𝐺₁₀∪𝐺₂₀=𝑀⧵𝑀+𝐺₁₀∪𝑀+𝐺₂₀=𝑀∩𝑀+𝐺₁₀∪𝑀+𝐺₂₀=𝑀∩𝑀+𝐺₁₀∩𝑀∩𝑀+𝐺₂₀=𝑀⧵𝑀+𝐺₁₀∩𝑀⧵𝑀+𝐺₂₀. (18)

Let 𝐺₁=𝐾₁, and let 𝐺₂=𝐾₂ in Proposition 6. We have Corollary 7.

Corollary 7. If 𝐾₁ and 𝐾₂ are cones, then 𝐸𝑀,𝐾₁∪𝐾₂=𝐸𝑀,𝐾₁∩𝐸𝑀,𝐾₂. (19)

In the following, we investigate some new properties of the efficient set when 𝑀 is the union, intersection, sum, or difference of two sets.

Lemma 8. If 𝐾 is a convex cone, then 𝐸𝑀₁⧵𝑀₂+𝐾,𝐾⊂𝐸𝑀₁,𝐾. (20)

Proof. If 𝐸[𝑀₁⧵(𝑀₂+𝐾),𝐾]=∅, the result obviously holds.
Otherwise, for all 𝑥∈𝐸[𝑀₁⧵(𝑀₂+𝐾),𝐾], if 𝑥∉𝐸(𝑀₁,𝐾), ∃𝑦₁∈𝑀₁,𝑘∈𝐾 such that 𝑥=𝑦₁+𝑘,𝑘≠0, where 𝑦₁∈𝑀₁=[𝑀₁⧵(𝑀₂+𝐾)]∪[𝑀₁∩(𝑀₂+𝐾)].
If 𝑦₁∈𝑀₁⧵(𝑀₂+𝐾), it contradicts that 𝑥∈𝐸[𝑀₁⧵(𝑀₂+𝐾),𝐾]. Hence 𝑦₁∈𝑀₁∩(𝑀₂+𝐾). ∃𝑦₂∈𝑀₂ such that 𝑦₁∈𝑦₂+𝐾, so 𝑥∈𝑦₁+𝐾⊂𝑦₂+𝐾⊂𝑀₂+𝐾, which still contradicts that 𝑥∈𝐸[𝑀₁⧵(𝑀₂+𝐾),𝐾]. In consequence, 𝑥∈𝐸(𝑀₁,𝐾).

A more accurate relationship between the two efficient sets in Lemma 8 is described in the following proposition.

Proposition 9. If 𝐾 is a convex cone, then 𝐸𝑀₁⧵𝑀₂+𝐾,𝐾=𝐸𝑀₁,𝐾⧵𝑀₂+𝐾. (21)

Proof. 𝐸[𝑀₁⧵(𝑀₂+𝐾),𝐾]⊂𝐸(𝑀₁,𝐾) obtained by Lemma 8.
Note that (𝑀₁⧵𝑀₂+𝐾)∩(𝑀₂+𝐾)=∅, 𝐸[𝑀₁⧵(𝑀₂+𝐾),𝐾]=𝐸[𝑀₁⧵(𝑀₂+𝐾),𝐾]∩(𝑀₂+𝐾). So 𝐸[𝑀₁⧵(𝑀₂+𝐾),𝐾]=𝐸[𝑀₁⧵(𝑀₂+𝐾),𝐾]∩(𝑀₂+𝐾)⊂𝐸(𝑀₁,𝐾)⧵(𝑀₂+𝐾).
Reciprocally, for all 𝑥∈𝐸(𝑀₁,𝐾)⧵(𝑀₂+𝐾), 𝑥∈𝐸(𝑀₁,𝐾), 𝑥∉𝑀₂+𝐾. If 𝑥∉𝐸[𝑀₁⧵(𝑀₂+𝐾),𝐾], ∃𝑦∈𝑀₁⧵(𝑀₂+𝐾) with 𝑥≠𝑦 such that 𝑥∈𝑦+𝐾, in contradiction to 𝑥∈𝐸(𝑀₁,𝐾). Therefore, 𝐸𝑀₁,𝐾⧵𝑀₂+𝐾⊂𝐸𝑀₁⧵𝑀₂+𝐾,𝐾. (22)

Usually 𝐸(𝑀₁⧵𝑀₂,𝐾)≠𝐸(𝑀₁,𝐾)⧵𝑀₂, is what Example 10 shows.

Example 10. Consider the following:𝑀₁=𝑥,𝑦∈𝑅²,𝑀∣0≤𝑥≤1,0≤𝑦≤1₂=𝑥,𝑦∈𝑅²,𝑀∣0≤𝑥≤1,0≤𝑦≤1−𝑥₁⧵𝑀₂=𝑥,𝑦∈𝑅²,𝐸𝑀∣0≤𝑥≤1,1−𝑥≤𝑦≤1₁,𝑅⁺₂=,𝐸𝑀{(0,0)}₁⧵𝑀₂,𝑅⁺₂=𝑥,𝑦∣𝑥+𝑦=1,0≤𝑥≤1,0≤𝑦≤1. (23)
About the efficient point set of differences between two sets 𝑀₁ and 𝑀₂, Proposition 11 gives the conclusion, without requiring 𝐾 to be a convex but a cone.

Proposition 11. Consider the following:𝐸𝑀₁⧵𝑀₂,𝐾⊇𝐸𝑀₁,𝐾⧵𝑀₂∪E𝑀₂,𝐾⧵𝑀₁ (24)

Proof. We have𝐸𝑀₁⧵𝑀₂,𝐾=𝑀₁⧵𝑀₂⧵𝑀₁⧵𝑀₂+𝐾₀=𝑀₁∩𝑀₂∩𝑀₁∩𝑀₂+𝐾₀⊇𝑀₁∩𝑀₂∩𝑀₁+𝐾₀∩𝑀₂+𝐾₀=𝑀₁∩𝑀₂∩𝑀₁+𝐾₀∪𝑀₂+𝐾₀=𝑀₁∩𝑀₂∩𝑀₁+𝐾₀∪𝑀₁∩𝑀₂∩𝑀₂+𝐾₀=𝐸𝑀₁,𝐾∩𝑀₂∪𝐸𝑀₂,𝐾∩𝑀₁=𝐸𝑀₁,𝐾⧵𝑀₂∪𝐸𝑀₂,𝐾⧵𝑀₁. (25)

Remark 12. In the previously mentioned proposition, if (𝑀₁∩𝑀₂)+𝐾₀=(𝑀₁+𝐾₀)∩(𝑀₂+𝐾₀), the equation holds.

Propositions 13 and 15 present the properties of the efficient point sets; when 𝑀 is the union or intersection of two sets, respectively, 𝐾 is not required to be a convex.

Proposition 13. Consider the following:𝐸𝑀₁∪𝑀₂,𝐾=𝐸𝑀₁,𝐾⧵𝑀₂+𝐾₀∪𝐸𝑀₂,𝐾⧵𝑀₁+𝐾₀. (26)

Proof. We have𝐸𝑀₁∪𝑀₂,𝐾=𝑀₁∪𝑀₂⧵𝑀₁∪𝑀₂+𝐾₀=𝑀₁∪𝑀₂∩𝑀₁∪𝑀₂+𝐾₀=𝑀₁∪𝑀₂∩𝑀₁+𝐾₀∪𝑀₂+𝐾₀=𝑀₁∪𝑀₂∩𝑀₁+𝐾₀∩𝑀₂+𝐾₀=𝑀₁∩𝑀₁+𝐾₀∩𝑀₂+𝐾₀∪𝑀₂∩𝑀₂+𝐾₀∩𝑀₁+𝐾₀=𝐸𝑀₁,𝐾⧵𝑀₂+𝐾₀∪𝐸𝑀₂,𝐾⧵𝑀₁+𝐾₀. (27)

Remark 14. It is obvious that if 𝐸(𝑀₁,𝐾)∩(𝑀₂+𝐾₀)=∅ and 𝐸(𝑀₂,𝐾)∩(𝑀₁+𝐾₀)=∅, then 𝐸𝑀₁∪𝑀₂,𝐾=𝐸𝑀₁,𝐾∪𝐸𝑀₂,𝐾. (28)

Proposition 15. Consider the following:𝐸𝑀₁∩𝑀₂,𝐾⊇𝐸𝑀₁,𝐾∩𝑀₂∪𝐸𝑀₂,𝐾∩𝑀₁. (29)

Proof. Notice that (𝑀₁∩𝑀₂)+𝐾₀⊆(𝑀₁+𝐾₀)∩(𝑀₂∩𝐾₀). 𝐸𝑀₁∩𝑀₂,𝐾=𝑀₁∩𝑀₂⧵𝑀₁∩𝑀₂+𝐾₀⊇𝑀₁∩𝑀₂⧵𝑀₁+𝐾₀∩𝑀₂+𝐾₀=𝑀₁∩𝑀₂∩𝑀₁+𝐾₀∩𝑀₂+𝐾₀=𝑀₁∩𝑀₂∩𝑀₁+𝐾₀∪𝑀₂+𝐾₀=𝑀₁∩𝑀₂∩𝑀₁+𝐾₀∪𝑀₁∩𝑀₂∩𝑀₂+𝐾₀=𝐸𝑀₁,𝐾∩𝑀₂∪𝐸𝑀₂,𝐾∩𝑀₁. (30)

Remark 16. (i) Usually the equality does not hold in Proposition 15. Let 𝑀₁={(𝑥,𝑦)∣0≤𝑥≤𝑦≤1}, 𝑀₂={(𝑥,𝑦)∣0≤𝑥−1/2≤𝑦≤1}. Then 𝐸𝑀₁∩𝑀₂=,𝐸𝑀,𝐾{(1/2,1/2)}₁,𝐾∩𝑀₂∪𝐸𝑀₂,𝐾∩𝑀₁=∅, (31)
(ii) for Proposition 15, if (𝑀₁∩𝑀₂)+𝐾₀=(𝑀₁+𝐾₀)∩(𝑀₂+𝐾₀), then 𝐸𝑀₁∩𝑀₂,𝐾=𝐸𝑀₁,𝐾∩𝑀₂∪𝐸𝑀₂,𝐾∩𝑀₁, (32) (iii) since 𝑀₁⧵𝑀₂=𝑀₁∩𝑀₂, Proposition 11 can be obtained by Proposition 15.

3. Efficiency of Four Most Representative DEA Models

For each of DEA models, Charnes et al. and Wei et al. (e.g., see [5, 7, 9]) establish an associative multiobjective linear programming model and prove that a DMU_𝑗0 is DEA efficient if and only if (𝑥_𝑗0,𝑦_𝑗0) (associating with the DMU_𝑗0) is a Pareto efficient solution of the multiobjective linear programming problem (for the details about DEA efficient of DMUs in DEA model, see [4, 5] and the references therein). This beautiful conclusion provides the multiobjective linear programming as an efficient tool to solve DEA problems. So the key point now is to find all efficient solutions of the multiobjective linear programming problems. This section offers a simple way to do this.

By the results of Section 2, in the following, we investigate the multiobjective linear programming problems corresponding to the four DEA models and develop the structures of the efficient point sets and the efficient solution sets of these programmings. Based on these, the relationships between the DEA efficiency of the four DEA models are obtained.

Denote 𝑥_𝑗=(𝑥_1𝑗,…,𝑥_𝑚𝑗)>0 (𝑥_𝑗∈𝑅^𝑚) to be the input vector for the 𝑗th decision making unit (DMU_𝑗), and 𝑦_𝑗=(𝑦_1𝑗,…,𝑦_𝑠𝑗)>0(𝑦_𝑗∈𝑅^𝑠), to be the output vector for the 𝑗th decision making unit, for 𝑗=1,…,𝑛. For convenience the notations 𝑥₀=𝑥_𝑗0,𝑦₀=𝑦_𝑗0, for 1≤𝑗₀≤𝑛 are given. Let 𝐶=𝑅^𝑚₊⊂𝑅^𝑚,  and let 𝐷=𝑅^𝑠₊⊂𝑅^𝑠. The orderings in 𝑅^𝑚 and 𝑅^𝑠 are defined by 𝐶 and 𝐷, respectively. 𝐾={(𝑐,𝑑)∣𝑐∈𝐶,𝑑∈𝐷},𝐾₁={(𝑐,−𝑑)𝑐∈𝐶,𝑑∈𝐷}, 𝐾₀=𝐾⧵{0}, and 𝐾₁₀=𝐾₁⧵{0. The production possibility set 𝑇,𝑇⊂{(𝑥,𝑦)∣𝑥∈𝑅^𝑚,𝑦∈𝑅^𝑠,𝑥≥0,𝑦≥0}, is based on postulate sets which are presented with a brief explanation (see [4, 5]). The four most representative models are, briefly, 𝐶²𝑅,𝐵𝐶²,𝐹𝐺, and 𝑆𝑇, which correspond to different production possibility sets 𝑇_𝐶²_𝑅, 𝑇_𝐵𝐶², 𝑇_𝐹𝐺, and 𝑇_𝑆𝑇, respectively, [4–10] 𝑇_𝐶²_𝑅=𝑥,𝑦∣^𝑛_𝑗=1𝑥_𝑗𝜆_𝑗≤𝑥,_𝑛𝑗=1𝑦_𝑗𝜆_𝑗≥𝑦≥0,𝜆_𝑗,𝑇≥0,𝑗=1,…,𝑛_𝐵𝐶2=𝑥,𝑦∣^𝑛_𝑗=1𝑥_𝑗𝜆_𝑗≤𝑥,^𝑛_𝑗=1𝑦_𝑗𝜆_𝑗≥𝑦≥0,_𝑛𝑗=1𝜆_𝑗=1,𝜆_𝑗,𝑇≥0,𝑗=1,…,𝑛_𝐹𝐺=𝑥,𝑦∣^𝑛_𝑗=1𝑥_𝑗𝜆_𝑗≤𝑥,^𝑛_𝑗=1𝑦_𝑗𝜆_𝑗≥𝑦≥0,_𝑛𝑗=1𝜆_𝑗≤1,𝜆_𝑗,𝑇≥0,𝑗=1,…,𝑛_𝑆𝑇=𝑥,𝑦∣^𝑛_𝑗=1𝑥_𝑗𝜆_𝑗≤𝑥,^𝑛_𝑗=1𝑦_𝑗𝜆_𝑗≥𝑦≥0,_𝑛𝑗=1𝜆_𝑗≥1,𝜆_𝑗≥0,𝑗=1,…,𝑛. (33)

Obviously, the following equalities and inclusions hold: (a)𝑇_𝐶²_𝑅=𝑇_𝐹𝐺∪𝑇_𝑆𝑇,(b)𝑇_𝐵𝐶²=𝑇_𝐹𝐺∩𝑇_𝑆𝑇,(c)𝑇_𝐶²_𝑅⊇𝑇_𝐹𝐺⊇𝑇_𝐵𝐶²,(d)𝑇_𝐶²_𝑅⊇𝑇_𝑆𝑇⊇𝑇_𝐵𝐶². (34)

For distinguishing the DEA efficiency of DMUs in DEA models, according to the equivalence of DEA efficiency in DEA models and Pareto efficiency in multiobjective linear programming obtained by Charnes et al. and Wei et al., we introduce the multiobjective linear programming problem corresponding to DEA models in the following. For the details about DEA models, see [7, 8].

Consider multiobjective linear programming problem: 𝑉-min𝑥,−𝑦,s.t.𝑥,𝑦∈𝑇. (𝑉𝑃) For 𝑇=𝑇_𝐶²_𝑅, 𝑇_𝐵𝐶², 𝑇_𝐹𝐺, or 𝑇_𝑆𝑇, the four multiobjective linear programming problems 𝑉𝑃_𝐶²_𝑅, 𝑉𝑃_𝐵𝐶², 𝑉𝑃_𝐹𝐺, and 𝑉𝑃_𝑆𝑇 correspond to the four DEA models respectively.

Definition 17. (𝑥₀,𝑦₀) is called an Pareto efficient (weak efficient) solution of (𝑉𝑃), if there is no (𝑥,𝑦)∈𝑇, (𝑥,𝑦)≠(𝑥₀,𝑦₀) such that 𝑥,−𝑦≤𝑥₀,−𝑦₀,𝑥,−𝑦<𝑥₀,−𝑦₀. (35)
Since 𝐾 defines the ordering in 𝑅^𝑚+𝑠(⊃𝑇), the previous inequality can be written as 𝑥₀,−𝑦₀∈𝑥,−𝑦+𝐾₀,𝑥₀,−𝑦₀∈𝑥,−𝑦+int𝐾. (36) When 𝐾 is a convex pointed cone, all conclusions in Section 2 hold for multiobjective programming problem (𝑉𝑃). Theorem 18 provides the equivalency of DEA efficiency of DMUs and the Pareto efficiency of the points corresponding to the DMUs. Consequently, the structures of the sets of efficient solutions to DEA models are obtained by solving the multiobjective programmings. Then the relationships between the DEA efficiencies of DMUs in DEA models are deduced.

Theorem 18 (see [5]). A DMU_𝑗0 is DEA efficient if and only if (𝑥₀,𝑦₀) is a Pareto efficient solution of (𝑉𝑃).

For convenience, we denote by DMU₀ DMU_𝑗0. In the following, we investigate the structures of the efficient point sets and the efficient solution sets of the four vector optimization problems. Let𝑀₀=𝑥_𝑗,−𝑦_𝑗,𝑇∣𝑗=1,…,𝑛₀=𝑥_𝑗,𝑦_𝑗,𝑀∣𝑗=1,…,𝑛_𝐶2𝑅=𝑥,−𝑦∣𝑥,𝑦∈𝑇_𝐶²_𝑅,𝑀_𝐵𝐶²=𝑥,−𝑦∣𝑥,𝑦∈𝑇_𝐵𝐶²,𝑀_𝐹𝐺=𝑥,−𝑦∣𝑥,𝑦∈𝑇_𝐹𝐺,𝑀_𝑆𝑇=𝑥,−𝑦∣𝑥,𝑦∈𝑇_𝑆𝑇. (37) The relations (𝑎), (𝑏), (𝑐), and (𝑑) imply: (𝑎)𝑀_𝐶²_𝑅=𝑀_𝐹𝐺∪𝑀_𝑆𝑇,(𝑏)𝑀_𝐵𝐶²=𝑀_𝐹𝐺∩𝑀_𝑆𝑇,(𝑐)𝑀_𝐶²_𝑅⊇𝑀_𝐹𝐺⊇𝑀_𝐵𝐶²,(𝑑)𝑀_𝐶²_𝑅⊇𝑀_𝑆𝑇⊇𝑀_𝐵𝐶². (38)

As an example, consider the multiobjective linear programming problem (𝑉𝑃_𝐶²_𝑅). Let 𝐸_𝐶²_𝑅 denote the efficient point set, and let 𝑆_𝐶²_𝑅 denote the efficient solution set. Note that 𝑥,−𝑦≤𝑥₀,−𝑦₀⟺𝑥₀,−𝑦₀∈𝑥,−𝑦+𝐾. (39) By Definitions 2 and 17, 𝐸_𝐶²_𝑅=𝑀₀⧵𝑀_𝐶²_𝑅+𝐾₀. (40) Since 𝑥₀,−𝑦₀∈𝑥,−𝑦+𝐾₀⟺𝑥₀,𝑦₀∈𝑥,𝑦+𝐾₁₀, (41) we have 𝑆_𝐶²_𝑅=𝑇₀⧵𝑇_𝐶²_𝑅+𝐾₁₀. (42) Similar argument is applied to other three multiobjective linear programming problems. Therefore the following two theorems hold.

Theorem 19. Consider the following:𝑎𝐸_𝐶²_𝑅=𝑀₀⧵𝑀_𝐶²_𝑅+𝐾₀,(𝑏)𝐸_𝐵𝐶²=𝑀₀⧵𝑀_𝐵𝐶²+𝐾₀,(𝑐)𝐸_𝐹𝐺=𝑀₀⧵𝑀_𝐹𝐺+𝐾₀,(𝑑)𝐸_𝑆𝑇=𝑀₀⧵𝑀_𝑆𝑇+𝐾₀. (43)

Theorem 20. Consider the following:(𝑎)𝑆_𝐶²_𝑅=𝑇₀⧵𝑇_𝐶²_𝑅+𝐾₁₀,(𝑏)𝑆_𝐵𝐶²=𝑇₀⧵𝑇_𝐵𝐶²+𝐾₁₀,(𝑐)𝑆_𝐹𝐺=𝑇₀⧵𝑇_𝐹𝐺+𝐾₁₀,(𝑑)𝑆_𝑆𝑇=𝑇₀⧵𝑇_𝑆𝑇+𝐾₁₀. (44)

Inclusions (𝑐), (𝑑) and Theorem 20 imply Theorem 21.

Theorem 21. Consider the following:(i)𝑆_𝐶²_𝑅⊆𝑆_𝐹𝐺⊆𝑆_𝐵𝐶²,(ii)𝑆_𝐶²_𝑅⊆𝑆_𝑆𝑇⊆𝑆_𝐵𝐶². (45)

Theorem 21 shows the relationships between the solutions of the four vector optimization problems. Therefore, by Theorem 18, the relationships between DEA efficiencies of DMUs in DEA models are as the following: (i) if DMU₀ is 𝐶²𝑅 DEA efficient, then it is 𝐹𝐺 DEA efficient. If 𝐷𝑀𝑈₀ is 𝐹𝐺 DEA efficient, then it is 𝐵²𝐶 DEA efficient;(ii) if DMU₀ is 𝐶²𝑅 DEA efficient, then it is 𝑆𝑇 DEA efficient. If 𝐷𝑀𝑈₀ is 𝑆𝑇 DEA efficient, then it is 𝐵²𝐶 DEA efficient.

By Proposition 13, along with the equality (𝑎) and (𝑎), we obtain following Theorem 22.

Theorem 22. Consider the following:𝐸_𝐶²_𝑅=𝑀₀⧵𝑀_𝐹𝐺+K₀⧵𝑀_𝑆𝑇+𝐾₀=𝑀₀⧵𝑀_𝑆𝑇+𝐾₀⧵𝑀_𝐹𝐺+𝐾₀=𝐸_𝐹𝐺⧵𝑀_𝑆𝑇+𝐾₀=𝐸_𝑆𝑇⧵𝑀_𝐹𝐺+𝐾₀. (46)

By Theorem 22, we conclude that if 𝐸_𝐶²_𝑅≠𝜙, then 𝐸_𝐹𝐺≠𝜙 and 𝐸_𝑆𝑇≠𝜙; by (𝑎), (𝑐), and (𝑑), we declaim that if 𝐷𝑀𝑈₀ is 𝐶²𝑅 DEA efficient, then it is also 𝐹𝐺 DEA efficient and 𝑆𝑇 DEA efficient.

Proposition 15 and (𝑏) infer Theorem 23.

Theorem 23. Consider the following:𝐸_𝐵𝐶²⊇𝐸_𝐹𝐺∩𝑀_𝑆𝑇∪𝐸_𝑆𝑇∩𝑀_𝐹𝐺. (47)

Consider that 𝐸_𝐹𝐺=𝑀₀⧵(𝑀_𝐹𝐺+𝐾₀), and 𝑀₀⊂𝑀_𝑆𝑇, if 𝐸_𝐹𝐺≠𝜙, then 𝐸_𝐹𝐺∩𝑀_𝑆𝑇≠𝜙. Similarly, 𝐸_𝑆𝑇∩𝑀_𝐹𝐺≠𝜙. Therefore, the following consequences are obtained:(i) if DMU₀ is 𝑆𝑇 DEA efficient, then it is also 𝐵𝐶² DEA efficient; (ii) if DMU₀ is 𝐹𝐺 DEA efficient, then it is also 𝐵𝐶² DEA efficient.

4. Conclusion

In this paper, Definition 2 presents a new representation to 𝐸(𝑀,𝐾). Then some new properties of 𝐸(𝑀,𝐾) are deduced by the new representation (Propositions 6 to 15 and their corollaries). Most of the properties are proved by mean of the binary operations of sets, which is a new method in multiobjective optimization. These conclusions are used to deal with the multiobjective linear programming problems corresponding to the four most representative DEA models. We investigate the structures of the efficient solution set of the four DEA models (Theorems 19 and 20) and deduce the relationships between DEA efficient solution sets of the four DEA models (Theorems 21 and 22). For each of the four DEA models, by Theorem 20, 𝑆=𝑇₀⧵(𝑇+𝐾₁₀); that is, all the DEA efficient DMUs are obtained by the binary operations of sets. Therefore, this is a simple way to get all DEA efficient DMUs theoretically. By using the new representation, it may be able to discuss the effects upon DEA efficiency of DMUs when the number of the DMUs changes in DEA models.

Acknowledgments

This research was supported in part by the Zhejiang Natural Science Foundation of China Grant Y6110054, the Shanxi Scholarship Council of China Research Grant 2010087.